Noncommutative Algebras

9970413我们的研究是为了更好地理解某些类的非交换生成的非交换代数的内部结构和表示理论。 具体的例子包括代数量子群和李超代数的包络代数。重点主要是在总理和原始理想理论，和根本的目标是研究原始谱在这种情况下，作为一种类型的非交换仿射代数几何。类似地，我们试图确定非交换Hopf代数和仿射代数群的原始谱之间的类比;后者的目标主要是由非交换Hopf代数在量子群理论中所起的作用所激发的。 我们所关注的技术问题包括态射、形变、链性、模扩张和“小”无限维量子群。我们的研究对象是非交换结合代数。 这些抽象的对象提供了一个有效的设置为研究系统的多项式方程的非交换变量，这样的方程是基本的二十世纪数学和物理。我们在这项研究中的目的是研究一种类型的“几何”所产生的某些非交换系统的方程，在类比的经典几何获得的解决方案，以多项式方程系统的交换变量。

9970413Our research is directed toward better understanding the internal structure and representation theory of certain classes of finitely generated noncommutative algebras. The specific motivating examples include algebraic quantum groups and enveloping algebras of Lie superalgebras. The focus is mainly on prime and primitive ideal theory, and the underlying goal is to study primitive spectra in this context as a type of ``noncommutative affine algebraic geometry.'' Similarly, we seek to determine analogies between the primitive spectra of noncommutative Hopf algebras and affine algebraic groups; this latter objective is largely motivated by the role noncommutative Hopf algebras play in the theory of quantum groups. The technical issues we are concered with in this work include morphisms, deformations, catenarity, module extensions, and ``small'' infinite dimensional quantum groups.The subjects of our research are noncommutative associative algebras. These abstract objects have provided an effective setting for studying systems of polynomial equations in noncommuting variables; such equations are basic to twentieth century mathematics and physics. Our aim in this research is to study a type of ``geometry'' arising from certain noncommutative systems of equations, in analogy to the classical geometry obtained from solutions to systems of polynomial equations in commuting variables.